I am an electrical engineer embarking on theoretical research in machine learning. I am studying Real and Functional analysis to give myself a good background and my question is related to that :
When I study about topological spaces, I read that in the definition of topology, we have following two assumptions :
a) Union of any collection of sets in the family belong to the family. b) Intersection of any finite collection of sets in the family belong to the family.
These definitions intrigue me and make me ask the question : What is special about finite vs. infinite sets with respect to operations of union and intersection that these subtle distinctions are made ? I often find it difficult to construct counter-examples to these scenarios in some cases. I want to understand the rationale behind these concepts.
Since, I am not a practiced mathematics major, I am looking for some advise about how to approach my studies ( books etc. ) so that I can properly study and understand these concepts.
In maths we can define things however we want, but what really matters is how useful a definition is. Typically definitions come from real examples. It is a process of looking at a concrete thing and figuring out the abstract rules that govern it. For example someone looked at integers $\mathbb{Z}$, rationals $\mathbb{Q}$, reals $\mathbb{R}$, finite modulo groups $\mathbb{Z}/n\mathbb{Z}$, symmetric group $S_n$ and so on, and realized that what they all have in common is that there is some binary operation $G\times G\to G$ that is associative, there is neutral element, and each element has an inverse. It is a useful abstraction, because now I can prove some things about an abstract group and it will apply to all concrete groups.
The same applies to topology. But here's a problem. Human mind looks for patterns, for symmetries, and when you see two axioms: one that works for all situations and the other that works only for finite cases, it bothers you. We don't like such asymmetries. And you are not alone, I sympathize with you. I'd rather see both axioms applying to all possible unions and intersections.
But there is a very simple reason for this asymmetry: this is how the reality works. And I encourage you to accept that and move on.
Let me elaborate a bit. Topology is an abstraction, that is supposed to capture well known properties of many different concrete models. So for example we often consider $\mathbb{R}$ with so called Euclidean topology, the one generated from the Euclidean metric (of course it is a separate question, why this topology/metric and not some other? But the answer is the same: because it is useful). And in that space an arbitrary union of open subsets is open. But that's not the case for intersections, e.g $\bigcap_{n=1}^{\infty}(-1/n,1/n)=\{0\}$ which is not open.
Note that there is a variant of topology where we assume that intersection of arbitrary family of open subsets is open. Also known as the Alexandrov topology. However it turns out to be a rather exotic theory, in practice it is rare to encounter a topological space that is Alexandrov. In fact a $T_1$ topological space is Alexandrov if and only if it is discrete. And overwhelming majority of spaces you will see in maths are $T_1$.